Theorem pnfinf

Description: Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018)

	Ref	Expression
Assertion	pnfinf	|- ( A e. RR+ -> A ( <<< ` RR*s ) +oo )

Proof

Step	Hyp	Ref	Expression
1		rpgt0	|- ( A e. RR+ -> 0 < A )
2		nnz	|- ( n e. NN -> n e. ZZ )
3	2	adantl	|- ( ( A e. RR+ /\ n e. NN ) -> n e. ZZ )
4		rpxr	|- ( A e. RR+ -> A e. RR* )
5	4	adantr	|- ( ( A e. RR+ /\ n e. NN ) -> A e. RR* )
6		xrsmulgzz	|- ( ( n e. ZZ /\ A e. RR* ) -> ( n ( .g ` RR*s ) A ) = ( n *e A ) )
7	3 5 6	syl2anc	|- ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) = ( n *e A ) )
8	3	zred	|- ( ( A e. RR+ /\ n e. NN ) -> n e. RR )
9		rpre	|- ( A e. RR+ -> A e. RR )
10	9	adantr	|- ( ( A e. RR+ /\ n e. NN ) -> A e. RR )
11		rexmul	|- ( ( n e. RR /\ A e. RR ) -> ( n *e A ) = ( n x. A ) )
12		remulcl	|- ( ( n e. RR /\ A e. RR ) -> ( n x. A ) e. RR )
13	11 12	eqeltrd	|- ( ( n e. RR /\ A e. RR ) -> ( n *e A ) e. RR )
14	8 10 13	syl2anc	|- ( ( A e. RR+ /\ n e. NN ) -> ( n *e A ) e. RR )
15	7 14	eqeltrd	|- ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) e. RR )
16		ltpnf	|- ( ( n ( .g ` RR*s ) A ) e. RR -> ( n ( .g ` RR*s ) A ) < +oo )
17	15 16	syl	|- ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) < +oo )
18	17	ralrimiva	|- ( A e. RR+ -> A. n e. NN ( n ( .g ` RR*s ) A ) < +oo )
19		xrsex	|- RR*s e. _V
20		pnfxr	|- +oo e. RR*
21		xrsbas	|- RR* = ( Base ` RR*s )
22		xrs0	|- 0 = ( 0g ` RR*s )
23		eqid	|- ( .g ` RR*s ) = ( .g ` RR*s )
24		xrslt	|- < = ( lt ` RR*s )
25	21 22 23 24	isinftm	|- ( ( RR*s e. _V /\ A e. RR* /\ +oo e. RR* ) -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )
26	19 20 25	mp3an13	|- ( A e. RR* -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )
27	4 26	syl	|- ( A e. RR+ -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )
28	1 18 27	mpbir2and	|- ( A e. RR+ -> A ( <<< ` RR*s ) +oo )